Self-locking Differential on a Formula Student Car

INTRODUCTION

Fig.1: MG06/11 at FSG 2011 event, Hockenheimring Germany

The purpose of this project is to simulate the advantages and the disadvantages on adopting a self-locking differential in the drivetrain system of the University of Padova Formula Student car of season 2011, the MG06/11.

Formula Student is a students engineering competition where student teams from around the world design, build, test, and race a small-scale formula style racing car. The cars are judged, by industry specialists, on a number of criteria in different typologies of events:

The circuits for the dynamic events are narrow, tortuous and not so fast in order to preserve the safety of students. In this situations the drive train is designed to optimize the traction of the car, to get out of the corners as fast as possible, and to increase the drivability and the handling of the car, to assist the driver during the race. For this reasons it’is logical to adopt a self-locking differential that responds quickly to the commands of the driver and delivers the maximum torque from the engine to the asphalt.

OBJECTIVES

In this analysis, the drive train adopted a self-locking differential with multi-plate clutches produced by “Drexler® Motorsport GmbH”, which is possible to see in the figure below:

Fig.2: Drexler® Formula Student limited slip differential V2 2010

In this analysis we wanted to compare the performance that is possible to get from a self-looking differential with the performance that is possible to get from a conventional differential.

The two different solutions were used in the same car with identical and real cinematic configuration of the suspensions. To compare the two different solutions we provided two different tracks with other two different conditions of asphalt in order to generate different grip conditions.

The two tracks are:

a curve with a constant radius of 10m to emphasize the maximum lateral acceleration in the condition of maximum torque generated by the engine;

a straight to estimate the maximum longitudinal acceleration and to simulate a real dynamic event in the Formula SAE called “acceleration”.

The two different conditions of asphalt are:

dry, to ensure the maximum grip between track and tires to put on the ground all the torque generated by the engine;

awash or freezing to ensure the minimum grip between the track and the tire and to underline the maximum torque ratio between the two half-shafts.

The Need for Locking Differential

The design of conventional differential gears has two important advantages:

the rotational speeds of the drive wheels can be adjusted independently to each other according to the different distances travelled by the left and right wheels and

the drive torque is symmetrically distributed to both drive wheels, without any yawing moment.

These two advantages are however offset by a serious disadvantage. When the frictional potential of the two drive wheels is different, the propulsive forces transmitted to the road surface for both drive wheels depends on the smaller frictional potential of the two. This comparison relates in this case to inner wheel compensation in the axle drive. This means, for example, that a wheel standing

on ice (or wet/oily asphalt) will spin and the other wheel standing on asphalt cannot transfer more torque than the one that is spinning. The vehicle therefore cannot move-off.

In order to overcome this disadvantage of conventional differential gears, the compensating action has to be inhibited in critical driving conditions. This can be effected by:

using self-locking differentials, also known as limited-slip or locking differentials. These are differentials with a compensating action that is deliberately tight and restricted. This enables them to transmit torque to one wheel even when the other wheel is spinning because of poor grip. This means losing the advantage of power transmission without yawing moment. The free adaptation of both wheel speeds to the different distances travelled by the two tracks is restricted. The axle shafts are more stressed because of the torque redistribution. Locking differentials are divided into load or torque controlled, and speed or slip controlled.

In this analysis the drive train adopts a self-locking differential with multi-plate clutches producted by “Drexler® Motorsport GmbH”. We have an exploded view of it, in which we can see the gears and the clutches.

The locking effect of a self-locking differential with multi-plate clutch relies on the torque-dependent internal friction generated in two multi-plate clutches mounted symmetrically in the differential cage. The self-locking action results from a combination of the load dependency and spring loading of the multi-plate clutches. The load-dependent locking effect (Fig. 3, at top) relies on the input torque T1 applied to the differential cage [1]being transmitted via the differential shaft [2]to two pressure rings [3]in the differential cage [1]that are torsionally locked but slide axially. Under load, locking forces arise automatically at the surfaces of the prism-shaped recesses [8]in the pressure rings (see detail in Fig. 3), pressing the clutch plates together. The outer plates [4]are torsionally locked to the differential cage [1], and the inner plates [5]are torsionally locked to the axle bevel gears [6].

The frictional contact between the plates thus opposes the different axle shaft speeds (for example when a wheel spins) with a precisely defined force. This effect increases as the input torque increases. Since the locking forces are proportional to the torque transmitted, the locking effect adapts to the changing engine torque and to the torque increase in the various gears, but the interlock value does not.

The plate springs [7] that can be fitted to preload the multi-plate clutch create a constant initial locking effect that is independent of the torque transmitted, but sometimes makes noticeable creaking noises. This makes the system capable of locking even on extremely unfavourable surfaces, for example one wheel on ice. There is nevertheless still the disadvantage that a differential of this type always has a basic locking torque. This can be undesirable when parking and when cornering without slip.

In the model adopted for the analysis of the behavior of the car, there isn’t the plate spring [7] because the track is more narrow, tortuous and not so fast. The presence of the constant locking effect (that is independent of the torque transmitted) gives to the car an over-steering behavior (no optimal condition). For this reason the plate spring was eliminated and replaced with a simple metal washer which has the same thickness of the loaded plate spring.

A further drawback that should be borne in mind is that during the self-locking or compensating process, the tooth geometry of the bevel gears changes adversely, because the friction clutches that have to be applied must not be free of clearance.

MODELING

MODELING OF THE DIFFERENTIAL

Fig.4: Exploded view of the differential

We already had the CAD files and the mass properties of all the differential parts, provided directly by the manufacturer.

The assembly follows fundamentally an axial disposition, so it was quite simple to dispose all the frictions and gears along the differential case, using mainly bracket joints with the desired degree of freedom disabled in the joint.On the outersplined frictions we also locked the rotation relatively to the differential body, in order to have a solid rotation of that parts, and unlocked that rotation on the innersplined frictions, so they could rotate freely.The same strategy was used for coupling the rotation of the side bevel gears to the innersplined frictions, leaving free the axial movement

Fig.5: The "C" structure used to link the gears

We have 2 side bevel gears and 4 bevel gear. On the model we used only 1 bevel gear, because on the real differential there are 4,simply for a matter of symmetry and load distribution, not for cinematic reason, which is the subject of this study.The gear joint works only with gears which rotation axis are fixed to the same ground part. So we created a “fictional” weightless part with a “C” shape in order to couple the three gears.

Another problem was the behavior of the frictions:

Fig.6: frictions contact element

Firstly we used a single pair of frictions per side, instead of the 4 contact surfaces per side present in the real system. As for the gears, it’s an expedient to distribute the forces and give enough torque-to-axial force ratio. We could create the same situation only by setting an adequate coefficient of friction.

We tried to use the CAD contact but the shape of the friction was too complicated to mesh properly, and LMS didn’t give the possibility to set many parameters.

The “friction” element worked only as an energy dissipation on a revolute joint, so we couldn’t make it depend on the axial force.

The solution was creating two spheres on one friction, and a plane on the other, and then a contact element between the spheres and the plane. We used an hertzian contact and set the Young modulus (206GPa) and Poisson’s ratio (0,3), to have little deformation of the spheres.

The optimum coefficient of friction to represent the real differential was found to be 0,8. We did some lock-up tests to see the maximum percent of lock.Obviously in the comparison tests with an open differential we set this parameter to 0 to avoid the lock-up.

Fig.7: ramp/axle contact element

The same kind of contact was used for the contact between the axle of the solar and the ramps of the differential.

A sphere was put on the edge of the axis, and a plane created on the surface of the ramp.

We used an hertzian contact and set the Young modulus (206GPa) and Poisson’s ratio (0,3) of steel, as it really is. We implemented both acceleration and brake ramps for both sides of the differential.

The plate springs can preload the multi-plate clutch, and create a constant initial locking effect that is independent of the torque transmitted. In this model of the differential the plate spring was substituted with a simple washer with the same axial thickness of plate springs in the normal condition of work into the differential cage and with the same mass properties of plate spring.

MODELING OF THE CAR

Fig.8: Model of the car

Once defined the differential we put it on the MG06/11, coupling it with a revolute joint on which apply the pinion torque. We yet had the CAD of the frame, body and the uprights. We could then create the points on which the suspension are linked, having the coordinates given by the “Lotus” optimization program.

Fig.9: Suspension model detail

We then created simple monodimensional elements for triangles, steer and suspension rods.Then we put TSDA elements where the real spring and damper are positioned.For the spring we used the real values of the basis setup of the car at the front and a quite harder suspension at the rear to emphasize the load transfer along the rear axis, to make the differential work more appreciably.

Ant

post

Free length [mm]

200

245

Spring [N/m]

40000

100000

Damping [kg/s]

100000

100000

For the tires we used the simple tire model with the parameters in the range recommended in the LMS online help. The only parameter we could verify was the friction coefficient of 1,2 given from the manufacturer as the maximum value in optimal condition of temperature.

Radius [mm]

255

Damping constant [kg/s]

3000

Rolling resistance

0.05

Friction coeff

1.2

Cornering stiffness [m kg s^-2 rad-1]

10000

Vertical stiffness [N/m]

300000

The Steering geometry was also the real one (respecting the Ackerman angle) and was controlled by a path follower control input and a control output acting with a force on the steer rod.

Other parts of the car (brake system, engine, exhaust system, driver, etc) were not implemented because not necessary in this analysis.

We creates a mass equal to the weight of the car and positioned where the center of mass of the vehicle is.

SIMULATIONS AND RESULTS

TEST ON THE STRAIGHT

Firstly we decided to make some tests on a straight to validate the differential model.

We created a simple straight path and used a path follower control input to maintain the vehicle on the road, using the following parameters for the control input:

Parameter

Value

Position gain

4

Velocity gain

400

Look ahead distance [mm]

3500

We used this control to generate an actuator force on the steer rod with a joint control output.

The test consisted in a period of assessment of 1 second to allowing the vehicle to gain a neutral position, and on the application of a torque curve to the differential until 6 seconds of simulation.

3 Different tests were made:

P1: WITH LOCK-UP, FULL GRIP

The first test was made with the tires at full grip. The car accelerates at its maximum capability without losing traction, the tires delivers to the ground all the torque arriving from the differential in equal parts.

The second test was made giving grip only to the right tires, and thus simulating a situation where left tires are on ice, to emphasize the differential work.

The coefficient of friction of the differential frictions was set to 0,8, the value we obtained after some tests verifying the manufacturer’s prescribed percent of lock-up for our differential configuration.

The car struggles to move, but even if the left tire is without grip, the locking of the differential allows the torque to reach the tire with grip, making the car accelerate.

With “open” differential the torque to right and left wheels is quite the same. With self-locking differential the most of the torque goes to the wheel with good grip, determinig a better acceleration.

Graph 2: Normal force on friction plates

The normal force on friction plates depends only on the applied torque and not on the locking percentage. Infact it’s quite the same with “open” and self-locking configuration.

Graph 3: Rotational speed of rear wheels

With full grip (P1) the wheels rotates at the same speed because they receive the same amount of torque.

In P2 the left wheel (with no grip) spins and the right one (with good grip) accelerates the car.

In P3 the left wheel spins much more consistently, taking away torque from the right wheel, that infact is slower than in P2, and so giving a worse acceleration to the car.

Graph 4: Real wheels slip

The Slip is defined as the difference between the speed of the drive wheel and a rear wheel divided by the speed of the drive wheel. This ratio highlights the numeric value between the speed of the differential cage and the rear wheels. With full grip (P1) there is no slip, with poor grip on left tires there is slip, decreasing with lock-up (P2) and constant with “open” differential (P3).

Graph 5: Distance covered by the car

Analysing P2 and P3 it’s possible to see how with the self-locking differential the car travels a bigger distance, because having a better acceleration.

Of course in P1 (having more grip) the car travels even more distance.

Graph 6: Vehicle speed

With the self-locking differential the car reaches higher velocities in the same time (6s) of simulation

TEST ON CURVE WITH CONSTANT RADIUS

In the second timewe decided to make some tests on a track with curve that has composed by a straight of 3.5m, a curve with constant radius of 10m and a angle of 90° and a finally straight of 5m.

This time we used the following parameters for the control input:

Parameter

Value

Position gain

20

Velocity gain

400

Look ahead distance [mm]

3500

The “path follower control input” generates an actuator force on the steer rod with a joint control output that give to the car the possible to follow the centerline of the path.

As in the case of the straight, the test consisted in a period of assessment of 1 second in order to allow the vehicle to gain a neutral position. The test ended at 5,7s.

We applied a torque curve to the body of the differential with a maximum of 420Nm.

P4: WITH LOCK-UP, FULL GRIP

The first test was made with the tires at full grip and, as seen in the straight test, the coefficient of friction of the differential frictions set to 0,8. The car accelerates at its maximum capability without losing the centerline of the path, but the tires haven’t the possibility to deliver to the ground all the torque arriving from the differential in equal parts. It’s possible to see in the torque graph that the maximum torque ratio between internal and external rear wheels is about 60%. That confirms the real value of torque ratio given by the producer of the differential, as expected.

In P5 the torque acting on the two half-shafts results identical due to the “open” differential. However in P4, the torque results different due to the self locking differential. As seen on the straight test, the most of the torque goes to the wheel with good grip. It’s possible to see that the maximum torque ratio between internal and external rear wheels is about 60% at its maximum, as expected.

Graph 8: Normal force on friction paltes

The graph shows the presence of high gradient of force when the torque quickly increases and when the torque slowly decreases (due to the angle of 50° on the ramp in deceleration).As in the straight the graphs are identical at the beginning and then differ because of the different behavior of the car at the exit of the corner.

Graph 9: Rotational speed of rear wheels

In P5 the angular velocity of the rear left wheel (internal wheel) is very high due to the loss of grip and the high amount of torque arriving from the differential. In P4 the angular velocity of the rear wheels are more similar due to the lock-up effect.

Graph 10: Real wheels slip

The Slip is defined as the difference between the speed of the drive wheel and a rear wheel divided by the speed of the drive wheel. This ratio highlights the numeric value between the speed of the differential cage and the rear wheels.

Bigger slips are reached with the “open” configuration of the differential (P5)

Graph 11: Yaw momentum

The immediate consequence of the differential lock, is the increase of yaw moment, due to the difference of the longitudinal forces of the rear wheels.

The car has much oversteer, requiring more driving skills from the driver, but determining better performances.

Graph 12: Arm of the equivalent thrust force

The arm of the equivalent thrust force is obtained diving the yaw momentum by the sum of the thrust forces on rear wheels. It’s positive if the force is on the right of the center of gravity of the car, so that the Momentum results positive.

A bigger arm results in a bigger yaw momentum.

Graph 13: Distance covered by the car

With the self-locking differential the maximum traveled distance is 20.3m, and with the “open” differential it’s only 19.5m for the same time of simulation.

Graph 14: Vehicle speed

The velocity of the car is exactly the same during the straight and the corner entrance, but when giving power at the corner apex, the self-locking differential gives better traction and permits a better acceleration out of the corner.

The speed at the and of the simulation is 10.6m/s with lock-up and only 9.6m/s with the “open” differential”

The advantages that a Self-Locking Differential can give in terms of maximum velocity and maximum distance travelled are at the detriment of the stability and drivability of the car, giving an oversteering behavior (instead of understeering behavior) with more effort and concentration required from the driver.

CONCLUSIONS

The tests on the straight permitted to validate the behavior of the LMS differential model, in particular the maximum achievable locking percentage. After setting the correct coefficient of friction for the friction plates, we saw that for different engine torques, the locking percentage was always about 60%, as prescribed by the manufacturer for the ramp we used in the simulation (40° inclination).

A test with left tires without grip (as on ice) showed the different behavior between open differential, with the car struggling to move forward and delivering most of the torque to the wheel with poor grip, and self-locking differential, with the car having less difficulties in moving because giving 60% of the torque to the wheel with good grip.

A second test with the car in a real race condition (a 90° corner afforded in full acceleration) showed the potential of the differential on the race track.

Doing a corner with the same curve of torque coming from the engine gave better results with the self-locking differential. The car covered more distance in the same time(20,4m vs 19,6m), also coming off the corner with higher speed (38,22km/h vs 34,74km/h).

The advantages are emphasized by the choice of parameters (corner radius, engine torque curve, tire grip) we made, but even if this advantage might be smaller, we have to consider that in a lap made of about 10-15 corners the advantage will be consistent and for sure it could make the difference.

Obviously a differential should be correctly set with the correct differential ramp (also in relation to the track) in order to avoid drivability issues, or poor agility in very sharp corners. A bad setting could result in worse performances comparing to an open differential, but this was not our focus.

In this work we validated the correct functioning of the differential model we implemented, and then we gave some examples of work conditions in which it demonstrated to work as expected, and also better than a common open differential if correctly set-up.